Optical Signatures of Band Flatness and Anisotropic Quantum Geometry in Magic-Angle Twisted Bilayer Graphene

We study the degree of band flatness and the anisotropic quantum geometry in magic-angle twisted bilayer graphene by varying the twist angle and the parameters of lattice relaxation using optical conductivity. We show that the degree of band flatness and its quantum geometry can be revealed through optical absorption and its resulting optical bounds, which are based on the trace condition in quantum geometry. More specifically, the narrow and isolated peak of optical absorption in the low-energy region provides information about the bandwidth of the two flat bands. When this value is smaller than the electron interaction, it serves as a critical condition for the emergence of flat band superconductivity. Furthermore, optical absorption also provides the gap value between the flat band and the dispersive band, and when this gap is larger than the electron interaction, it facilitates the realization of fractional Chern insulating phases. We show that the narrow and isolated peak of optical bound near zero energy decreases as lattice relaxation increases. Meanwhile, we demonstrate that the imaginary part of (generalized) optical Hall conductivity reveals the vanishing of the negative part of Berry curvature, which is enforced by the refined trace-determinant inequality. Accordingly, we show that the total amount of the negative part and component of the Berry curvature approaches zero in the single ideal flat-band case. In contrast, when considering all occupied bands, the total amount of the negative component is slightly different from zero. Lastly, we demonstrate that the vanishing of flat band velocity and the emergent chiral symmetry are sufficient conditions for the saturation of the trace condition, which pertains to the isotropic case. In contrast, the determinant condition can only be saturated in anisotropic systems.